We firstly discuss the geometric phase rotation for an electromagnetic wave traveling along the optical fiber in Minkowski space. We define two types of novel geometric phases associated with the evolution of the polarization vectors in the normal and binormal directions along the optical fiber by identifying the normal-Rytov parallel transportation law and binormal-Rytov parallel transportation law and derive their relationships with the new types of Fermi-Walker transportation law in Minkowski space. Then we describe a novel approach of solving Maxwell's equations in terms of electromagnetic field vectors and geometric quantities associated with the curved path characterizing the path uniform optical fiber by using the traveling wave transformation method. Finally, we investigate that electromagnetic wave propagation along the uniform optical fiber admits an interesting family of Maxwellian evolution equation having numerous physical and geometric applications for anholonomic coordinate system in Minkowski space

In this paper, we relate the evolution equations of the electric field and magnetic field vectors of the polarized light ray traveling in a coiled optical fiber in the ordinary space into the nonlinear Schrödinger’s equation by proposing new kinds of binormal motions and new kinds of Hasimoto functions in addition to commonly known formula of the binormal motion and Hasimoto function. All these operations have been conducted by using the orthonormal frame of Bishop equations that is defined along with the coiled optical fiber. We also propose perturbed solutions of the nonlinear Schrödinger’s evolution equation that governs the propagation of solitons through the electric field [Formula: see text] and magnetic field [Formula: see text] vectors. Finally, we provide some numerical simulations to supplement the analytical outcomes.

In this paper, we relate the evolution equation of the electric field and magnetic field vectors of the polarized light ray traveling along with a coiled optical fiber on the unit 2-sphere S² into the nonlinear Schrödinger's equation by proposing new kinds of binormal motions and new kinds of Hasimoto functions in addition to commonly known formula of the binormal motion and Hasimoto function. All these operations have been conducted by using the orthonormal frame of spherical equations that is defined along with the coiled optical fiber lying on the unit 2-sphere S². We also propose perturbed solutions of the nonlinear Schrödinger's evolution equation that governs the propagation of solitons through electric field (E) and magnetic field (M) vectors. Finally, we provide some numerical simulations to supplement the analytical outcomes.

In this study, we investigate the special type of magnetic trajectories associated with a magnetic field [Formula: see text] defined on a 3D Riemannian manifold. First, we consider a moving charged particle under the action of a frictional force, [Formula: see text], in the magnetic field [Formula: see text]. Then, we assume that trajectories of the particle associated with the magnetic field [Formula: see text] correspond to frictional magnetic curves ([Formula: see text]-magnetic curves[Formula: see text] of magnetic vector field [Formula: see text] on the 3D Riemannian manifold. Thus, we are able to investigate some geometrical properties and physical consequences of the particle under the action of frictional force in the magnetic field [Formula: see text] on the 3D Riemannian manifold.

In this paper, we study a special type of magnetic trajectories associated with a magnetic field [Formula: see text] defined on a 3D Riemannian manifold. First, we assume that we have a moving charged particle which is supposed to be under the action of a gravitational force [Formula: see text] in the magnetic field [Formula: see text] on the 3D Riemannian manifold. Then, we determine trajectories of the charged particle associated with the magnetic field [Formula: see text] and we define gravitational magnetic curves ([Formula: see text]-magnetic curves) of the magnetic vector field [Formula: see text] on the 3D Riemannian manifold. Finally, we investigate some geometrical and physical features of the moving charged particle corresponding to the [Formula: see text]-magnetic curve. Namely, we compute the energy, magnetic force, and uniformity of the [Formula: see text]-magnetic curve.

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